# Conditional and absolute convergence

• Nov 4th 2010, 10:31 AM
Newtonian
Conditional and absolute convergence
For each $y \in (0,1]$, find a series of the form $\sum\limits_{n=1}^{\infty}{a_nn^{-x}}$ which is absolutely convergent for $x>1$ but not for $x<1$ and is (conditionally) convergent for $x>y$ but not for $x.
• Nov 4th 2010, 02:50 PM
Drexel28
Quote:

Originally Posted by Newtonian
For each $y \in (0,1]$, find a series of the form $\sum\limits_{n=1}^{\infty}{a_nn^{-x}}$ which is absolutely convergent for $x>1$ but not for $x<1$ and is (conditionally) convergent for $x>y$ but not for $x.

Is the sequence supposed to depend on $y$?
• Nov 4th 2010, 03:07 PM
Newtonian
Yes; we may assume we're given a $y \in (0,1]$ and we want to find a series as stated (by defining $a_n$, which are allowed to depend on the given $y$).